Trigonometry Examples

$9 \csc^{4} (z) - 6 \csc^{2} (z) + 1$

Step 1

Rewrite $\csc^{4} (z)$ as ${(\csc^{2} (z))}^{2}$ .

$9 {(\csc^{2} (z))}^{2} - 6 \csc^{2} (z) + 1$

Step 2

Let $u = \csc^{2} (z)$ . Substitute $u$ for all occurrences of $\csc^{2} (z)$ .

$9 u^{2} - 6 u + 1$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $9 u^{2}$ as ${(3 u)}^{2}$ .

${(3 u)}^{2} - 6 u + 1$

Step 3.2

Rewrite $1$ as $1^{2}$ .

${(3 u)}^{2} - 6 u + 1^{2}$

Step 3.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 u = 2 \cdot (3 u) \cdot 1$

Step 3.4

Rewrite the polynomial.

${(3 u)}^{2} - 2 \cdot (3 u) \cdot 1 + 1^{2}$

Step 3.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 3 u$ and $b = 1$ .

${(3 u - 1)}^{2}$

Step 4

Replace all occurrences of $u$ with $\csc^{2} (z)$ .

${(3 \csc^{2} (z) - 1)}^{2}$